# change of order of integration

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##### 1: Bibliography L

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Changing the order of integration.
J. Austral. Math. Soc. 11, pp. 421–432.
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Addendum to: “Changing the order of integration”.
J. Austral. Math. Soc. 14, pp. 383–384.
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##### 2: 1.5 Calculus of Two or More Variables

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###### Change of Order of Integration

…##### 3: 19.28 Integrals of Elliptic Integrals

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##### 4: 2.6 Distributional Methods

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►Motivated by Watson’s lemma (§2.3(ii)), we substitute (2.6.2) in (2.6.1), and integrate term by term.
…Inserting (2.6.2) into (2.6.1) and integrating formally term-by-term, we obtain
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►The

*Stieltjes transform*of $f(t)$ is defined by … ►On substituting (2.6.15) into (2.6.26) and interchanging the order of integration, the right-hand side of (2.6.26) becomes … ►In terms of the*convolution product*…##### 5: 1.17 Integral and Series Representations of the Dirac Delta

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►Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)):
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►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these

*non-${L}^{2}$ improper eigenfunctions*of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. … ►Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …##### 6: 2.10 Sums and Sequences

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►Assume that $a,m$, and $n$ are integers such that $n>a$, $m>0$, and ${f}^{(2m)}(x)$ is absolutely integrable over $[a,n]$.
Then
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►This identity can be used to find asymptotic approximations for large $n$ when the factor ${v}_{j}$
changes slowly with $j$, and ${u}_{j}$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i).
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(b´)
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►In these circumstances the integrals in (2.10.28) are integrable by parts $m$ times, yielding
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On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity ${z}_{j}$, say,

2.10.30
$$f(z)-g(z)=O\left({(z-{z}_{j})}^{{\sigma}_{j}-1}\right),$$
$z\to {z}_{j}$,

where ${\sigma}_{j}$ is a positive constant.

##### 7: Errata

##### 8: Philip J. Davis

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►An interesting anecdote told by Davis reveals that he and mathematician Philip Rabinowitz were dubbed “Heroes of the SEAC” when their Gaussian integration code executed correctly on its first run.
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►Davis also co-authored a second Chapter, “Numerical Interpolation, Differentiation, and Integration” with Ivan Polonsky.
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►Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958.
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►The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling.
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##### 9: 2.5 Mellin Transform Methods

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###### §2.5(i) Introduction

… ►We now apply (2.5.5) with $$, and then translate the integration contour to the right. … ►Let $f(t)$ and $h(t)$ be locally integrable on $(0,\mathrm{\infty})$ and …Also, let … ►(The last order estimate follows from the Riemann–Lebesgue lemma, §1.8(i).) …##### 10: 2.3 Integrals of a Real Variable

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